MAT 145: Test #4 (50 points)
Part 2: Calculator OK!
Name ________________________ Calculator Used ____________
3
22. Calculate the exact value for the Riemann Sum approximation of
Score
1
∫ 2x
2
____________________
dx using R4. Sketch a graph that
1
includes your rectangles, organize your calculations, and express R4 as a common fraction reduced to
lowest terms. (4 pts)
_______________________
x
23. For g(x) =
∫
1
5t 3 − sin 2t dt , use the Fundamental Theorem of Calculus to determine g !(x) . (2 pts)
( )
_______________________
24. Suppose that k(x) is the cost of producing coaxial cable, given in dollars per meter, with k(x) =
3+ 20x
2x +1
50
for x ≥ 0. Explain the meaning and interpretation of
∫ k(x) dx = 981.44 for this context. (2 pts)
0
dV
= 1+ e −t liters/day, where V is
dt
the volume in liters and t is the time in days from the instant evaporation was first measured. Create
and evaluate a definite integral to determine how much water has evaporated from this pond in
the first 10 days that evaporation was measured. (3 pts)
25. The rate at which water is evaporating from a pond is modeled by
_______________________
5
26. Lance walked along a straight path beginning at point P.
A graph of Lance’s velocity is shown here, in miles per hour.
After 9 hours of walking, how far was Lance from the starting
point P? (2 pts)
4
V
3
2
1
________________
–4
t
–2
2
4
6
8
10
–1
–2
–3
–4
27. After World War II, the birth rate in western countries increased dramatically. One model for that birth
rate, during the first 10 years after the war, is given by b(t) = 5+ 2t, 0 ≤ t ≤ 10 , in millions of babies per year.
At what time, call it T0, stated in years, did the total number of births reach 14 million? (2 pts)
_______________________
12
Name
___________________________________________________________________
BONUS!
BONUS!
BONUS!
(A) The rate of sales of a product is given by S(t) = 25e kt , where t is the time in years with t = 0
corresponding to January 1, 1990, and S is measured in thousands of units per year. The value k is a
constant. Explain your responses and show calculus evidence to justify your solutions.
a. What was the sales rate at the beginning of 1990? (1 pt)
b. Calculate the average sales rate for the 4-year period beginning January 1, 1994. (2 pts)
c. Determine the exact value of the constant k that leads to the sales rate doubling every 8 years.
(2 pts)
(B) Population density measures the number of people per square mile that inhabit a given living area.
For a certain city, the population density decreases as you move away from the center of the city. The
population density at a distance r miles from the city center can be modeled by the function
D(r) = 10000 2 − r . Calculate the total population of the city. Explain your calculation and show
(
)
calculus evidence to justify your solution. (3 pts)
(C) Oil is leaking from a storage tank at the rate R(t) = 4000e −0.3t liters/day. Suppose that the leak is
never fixed. Determine the total amount of oil lost. Explain your calculations and show calculus
evidence to justify your solution. (2 pts)
Calculus I
MAT 145
Test #4: 50 points
Evaluation Criteria
Part I: No Calculators (35 points)
(1) – (10): 1 pt each; no partial credit
(11) – (12): 2 pts each; partial credit possible; simplify where appropriate
(13) – (16): 2 pts each set of four options; no partial credit
(17): 3 pts: state exact x-axis interval, show evidence for your response, and write a sentence to connect your
evidence and solution
(18): 3 pts: state exact values of x, show evidence for your response, and write a sentence to connect your
evidence and solution
(19): 3 pts: state exact values of x, show evidence for your response, and write a sentence to connect your
evidence and solution
(20) – (21): 2 pt each; partial credit possible; simplify where appropriate
Part II: Calculators May Be Used (15 points)
(22) 4 pts: show sketch of function graph and correct rectangles, organize your calculations, and show
requested Riemann sum as a common fraction reduced to lowest terms
(23) 2 pts: show correct expression for g !(x)
(24) 2 pts: accurate and clear explanation including use of appropriate unit labels
(25) 3 pts: correct definite integral; correct calculation with appropriate indication of units
(26) 2 pts: accurate response with appropriate evidence and label
(27) 2 pts: accurate response with appropriate calculus-based evidence and correct label
BONUS! Provide complete and accurate solutions with justifications.
(A) 5 pts: (a) 1 pt: correct and correctly labeled sales rate; (b) 2 pts: accurate determination of value for k
with appropriate evidence; (c) 2 pts: accurate determination of average value with appropriate
calculus evidence, labeled appropriately
(B) 3 pts: Correct calculation with appropriate evidence and explanation
(C) 2 pts: Correct calculation with appropriate evidence and explanation